CHAPTER 2.

Tight Binding Calculation of Molecules and Solids

In carbon materials except for diamond, the 11" electrons are va­

lence electrons which are relevant for the transport and other solid

state properties. A tight binding calculation for the 1r electrons is

simple but provides important insights for understanding the elec­

tronic structure of the'll" energy levels or bands for graphite and

graphite-related materials.

2.1 Tight Binding Method for B Crystalline Solid

In this section we explain the tight binding method for a crystalline solid. In the

following sections, we show some examples of energy bands for carbon materials

discussed in the Chapter 1.

!.1.1 Secular Equation

Because of the translational symmetry of the unit cells in the direction of the

lattice vectors, ai, (i = 1,"',3), any wave function of the lattice, lIf, shouldsatisfy Bloch's theorem

(i=I,···,3), (2.1)

where Til; is a translational operation along the lattice vector ai, and k is the

wave vector[45, 46J. There are many possible functional forms of lit which satisfy

Eq. (2.1). The most commonly used form for 11' is a linear combination ofplane waves. The reason why plane waves are commonly used is that: (1) the

integration of the plane wave wavefunction is easy and can be done analytically,

(2) the numerical accuracy only depends on the number of the plane waves

uaed. However, the plane wave method also has limitations: (1) the scale of the

17

18 CHAPTER 2. TIGHT BINDTNG CALCULATION

computation is large, and (2) it is difficult to relate the plane wave wavefuoclionto the atomic orbitals in the solid.

Another functional form which satisfies Eq. (2.1) is based on the j-th atomic

orbital in the unit cell (or atom). A tight binding, Bloch function ~i(f, rj is

given by,_ 1 N ._ _ _

<l>j(k,'1= ,..L>··R~j(f-R), (j=I,···,n). (2.2)vN A

Here Ii. is the position of the atom and "Pj denotes the atomic wavefunction in

state j. The number of atomic wavefunctions in the unit cell is denoted by fl,

and we have n Bloch functions in the solid for a given k. To form ~j(f, r) in

Eq. (2.2), the 'f'j's in the N (- 1024) unit cells are weighted by the phase factor

exp(ik. R) and are then summed over the lattice vectors if of the whole crystal.

The merits of using atomic orbitals in Bloch functions are as follows: (1) the

/lumber of basis functions, n, can be small compared with the number of plane

waves, and (2) we can easily derive the formulae for many physical properties

using this method: Hereafter wecollsider the tight binding functions of Eq. (2.2)

to represent the Block fundions.

It is clear that Eq. (2.2) satisfies Eq. (2.1) since

N

~ (" -+-) 1" ,'·A (- - R-)~j A",r a = NL.....e !Pj r+a-A

.,.• 1 .f!-. .'·(A-.) (_ (R- _)) (2.3)e NL.....e !pjr--aA_.

= e,t·.f4t·(f T'J ,'1>

where we use the periodic boundary condition for the M =- N- 1/ 3 unit vectors

in each Go direction,

<l>j(k,f+Mii;) = <l>j(k,'1 (i= 1,···.3), (2.4)

consistent with the boundary condition imposed on the translation vector TM.f; =I. From this boundary condition, the phase factor appearing in Eq. (2.2) satis­

fies exp{ikMa,} = 1, from which the wave number k is related by the integer

-The fun.i14t.ions of the tj~t bin~mabod are that.: (1) th_ it: no umple rule to improvethe nwnerical accuracy and (2) atomic ori>itaJ.. do not. deKribe the interatomic repon.

2.1. TIGHT BINDlNG METHOD FOR A CRYSTALLINE SOLID 19

p,

k= 2,.,., (p=O,I ..... M-I), (i=I, ... ,3).MOi

(2.5)

In three dimensions, the waveved.or r is defined for the z, y and z directions,

as .1:%. k, and k~. Thus M 3 = N wave vectors exist in the first Brillouin zone,where the .1:; CaD be considered as continuum variables.

The eigenfunctions in the solid ~ j(r, r') (j = I, . ",0), where n is the numberof Bloch wavefunctions, are expressed by a linear combination of Bloch functions

¢.i'(k, r) as follows:

•"';(k, '1 = L: C;;o(k)",;-<k. '1.

;'=1(2.6)

where Cjj.(r) are coefficienLs to be determined. Since the functions 'i'j(f. r')should also satisfy Bloch's theorem, the summation in Eq. (2.6) is taken onlyfor tbe Bloch orbitals ~i'(f, i) with the same value of f.

The j-th eigenvalue Ej(k) (j = I", " n) as a function of f is given by

(2.7)

where 1{. is the Hamiltonian of the solid. Substituting Eq. (2.6) into Eq. (2.1)and making a change of subscripts, we obtain the following equation,

• •L: Ci;Cw("';11I1"'r} L: 1I;;,(k)Ci;Cw

E;(k) = jj':l == jj~=l

L: Ci;Cw{"';I"'r} L: S;;o(k)Ci;C;;,jj'=l jj'=l

(2.6)

where the integrals over the Bloch orbitals, 7ijj'(k) and SU,(k) are called transfer

integral matrices and overlap integral matrices, respectively, which are definedby

11;;4) = ("';1111"';')' Sir(k) = ("';1"';') U.i' = 1... · ••). (2.9)

When we fix the values of the n x n matrices 1f.jj,(k) and Sjj'(k) in Eq. (2.9)for a given k value, the coefficient Ctj is optimized so as to minimize E;(k).

20 CHAPTER 2. TIGHT BINDING CALCULATION

It is noted that the coefficient Gij is also a function of k, and therefore Cij is

determined for each k. When we take a partial derivative for Cij while fixing

the other Gil', Cir I and Ci; coefficients,r we obtain zero for the local minimumcondition as follows,

N

L 1/;;'(k)Cwj'=l

N

L Sji·(k)c;jCWjJ'=1

N

" 1/. ·,(k)C7.C··,L..J 11 ,,'J N

iJ'-1 "'--!-"-='--------,-" L S;p(k)CW = o.

(~, S;P(k1Ci;CW) p=.

(2.10)

(2.11)

N

When we multiply both sides of Eq. (2.10) by L:: Sjj'(k)CiJCjj' and llubstitutejJ'=1

the expression for E;(k) of Eq. (2.8) into the second term of Eq. (2.10), we obtain

N N

L 1/;;,(k)CW = E,(k) L S;;,(k)CW ·;'=1 ;'=1

Defining a column vector,

Eq. (2.11) is expressed by

1/C, = E,(k)SC,.

(2.12)

(2.13)

Transposing the right hand side of Eq. (2.13) to the len, we obtain (1l ­E;(k)S]C; = O. If the inverse of the matrix [1t - E;(k)S] exists, we multi­

ply both sides by [1£ - Ej(k)Sj-l to obt.ain Ci = 0 (where 0 denot.es t.he null

vector), which means that. no wavefunct.ion is obt.ained. Thus t.he eigenfunc­

tion is given only when the inverse mat.rix does Dot exist, consistent with t.he

condit.ion given by

d.,[1/ - ESJ =0, (2.14)

where Eq. (2.14) is called t.he secular equat.ion, and is an equation of degree

n, wh06e solution gives all n eigenvalues of E;(k) (i = l,···n) for a given k.'Since G;J i. senerally .. complu: yari.t>le .ith two devea or freedom, .. raJ and .. complexpan, both GiJ and C:j e&ll be yaried independe:nlJ,..

2.1. TIGHT BINDING METIIOD FOR A CRYSTALLINE SOLID 21

Using the expression for E;(k) in Eqs. (2.7) and (2.1l), the coefficients C; as a

function of k are determined. In order to obtain the ener~ dispersion relations

(or energy bands) Eo(;;), we solve the secular equation Eq. (2.14), for a number

of high symmetry k points.

t.l.t Procedure for obtaining the ener" di"persion

In the "ight binding method, the one-electron ener~ eigenvalues E;(k) are ob­tained by solving tbe secular equation Eq. (2.14). The eigenvalues E;(k) are

a periodic function in the reciprocal lattice, which can be described within thefirst Brillouin zone. In a two or three dimensional solid, it is difficult to show

the energy dispersion relations over the whole range of f values, and thus we

plot E;(f) along the high symmetry directions in the Brillouin 'tone. The actual

procedure of the tight binding calculation is as follows:

1. Specify the unit cell and the unit vectors, ii;. Specify the coordinates of

the atoms in the unit cell and select n atomic orbitals which are considered

in the calculation.

2. Specify the Brillouin zone and the reciprocal lattice vectors, hj • Select the

high symmetry directions in the Brillouin zone, and k points along the

high symmetry axes.

3. For the selected k points, calculate the transfer and the overlap matrix

element.,7-l;j and Sij."

4. For the selected k points, solve the secular equation, Eq. (2.14) and obtain

the eigenvalues E i ( k) (i = I," " n) and the coefficients Cu(f).

Tigb....binding calculations are not self-consistent calculations in which the

occupation of an electron in an energy band would be determined. self-consistently.

That is, for given electron occupation, the potential of tbe Hamiltonian is cal­

culated, from which the updated electron occupation is determined using, for

example, Mulliken's gross population analysis [47]. When the input and the

output of occupation of the electron are equal to each other within the desired

accuracy, the eigenvalues are said to have been obtained self-consisLeDtiy.

• When only the tI'alUler matrix is calculated and the overlap matrix is taken .. the unit matrix,then \he Slater-KOlter extrapolalion aeme ",ulta.

22 CHAPTER 2. T1GHT BINDING CALCULATION

: ':,

i7 !7, .: C : c

, )(A)'-!B))' " /c: c: cI! I ~ I

H ~ H! H:. .:

Fig. 2.1: The unit cell of traM­

polyacelylene bounded by a boxdefined by t.he dot.ted lines, andsbowin& two inequivalent carhonatoms, A and H, in the unit cell.

in applying these calculational approaches to real systems, the symmetry of

the problem is considered in detaiJ on the basis of a tight.-binding approach andthe transfer and the overlap matrix elements are onen treated as parameters se­

lected to reproduce the band structure of the solid obtained either experimentallyor from first principles calculations. Both extrapolation methods sucb as k· pperturbation theory or interpolation methods using the Sialer-Koster approach

are commonly employed for carhon-related systems such as a 20 graphene sheet

or 3D graphite [9].

2.2 Electronic Structure of Polyacetylene

A simple example of'll'-energy bands ror a one-dimensional carbon chain is poly·

acetylene (see Sect. 1.2.2). In Fig. 2.1 we show, within the box defined by thedotted lines, the unit cell ror trnns.polyacetylene (CII)... which contains two in·equivalent carbon atoms, A and B, in the unit cell. As discussed in Sect. 1.2.2,

there is one r-electron per carbon atom, thus giving rise to two 'lI'·energy bandscalled bonding and anti-bonding 1I"-bands in the first Brillouin zone.

The lattice unit vector and the reciprocal lattice vector or this one-dimensionalmolecule are given by al = (0,0,0) and hI = (0/211',0,0), respectively. The Bril­louin zone is the line segment -0/11" < k < air. The Bloch orbitals consistingor A and B atoms are given by

"j(r) = ;..L: ,"R·~j(r - 11,,), (0 =A, B)vN R..

(2.1')

where the summation is taken over the atom site coordinate Ro for the A or B

2.2. ELECTRONIC STRUCTURE OF POLYACETYLENE 23

(2.17)

carbon atoms in the solid.

The (2 x 2) mat.rix Hamiltonian, 1(,0" (a,p =- A,B) is obtained by 8ubsti-­

t.utin~ Eq. (2.15) into Eq. (2.9). When a =- P=- A,

'X••(r) = ~ L ,"(R-R'l(IP.(r - R')I'XIIP.(r - R»RR'

~ t <" +~ L ,""(IP.(r - R')I'XIIP.(r - R»R=R' R=R'to (2.16)

+(terms equal to or more distant than R =- Fe ± 20)

l2" + (terms equal to or more distant than R =- R! ± a).

In Eq. (2.16) the maximum contribution to the matrix element llAA comes from

R =- R I, and this gives the orbital energy of the 2p level, (2p" The next order

contribution to 1iAA comes from terms in R =- R! ±a, which will be negleded forsimplicity. Similarly, 1i.BB also gives lZp for the same order of approximation.

Next let us consider the matrix element 'HAB(r). The largest contributionto 'HAB(r) arises when atoms A and B are nearest neighbors. Thus, in thesummation over R!, we only consider the cases Fe :; R ± a/2 and neglect moredistant terms to obtain

'X••(r) = ~ L {'''·''(IP.(r - R)I1iIIP.(r - R - 0/2»R

+ ,-;""(IP.(r - R)I1iIIP.(r - R+ 0/2»}2. '06(ko/2)

where t.he t.ransfer integral t is t.he integral appearing in Eq. (2.17) and denotedby!.

(2.18)

It is stressed that t has a negative value. The matrix element 1is A (r) is obtainedfrom 1lAs(r) through the nermitian conjugation relation 'ltoA :::: ?lAD' but since'HAS is real, we obtain 1t.BA :::: 1lAB.

·Note that (2" i. not .imply the ~Lomie energy value for the free atom, becauM: the HlUlliltonillneontaina a cry.tal potential.tHere ....e ~ve M8umed that all the :It bonding orbit.aI. are equal (l.SA bond.). In the real(CH):r compound, bond alternation occurs, in ....hich the bond energy alternate. betwe.m 1.7Aand 1.3A bond_, and the two atomic intep-",iODA in Eq. (2.17) are not equal. AlthouYt them.tortion of LIle lattice lowen the entT!Y, Lhe electronic enersy alway_ decreues more thanthe Iattioe energy in a one-dimension.:! material, and th.... the lattia become. deformed by aprocea called the Peierl. instability. See detail. in Sect. 11.3.1

CHAPTER 2. TIGHT BINDING CALCULATION24

4.0

3.0

2.0

~ 1.0

0.0

-1.0

-2.0-1.0 .0.5 0.0

W.

E

0.5 1.0

Fig. 2.2: The energy dispersionrelation E±(k) for polyacetylene[(CH)zJ, given by Eq. (2.21) withvalues for the parameters t =-1and s = 0.2. Curves E+(k) andE_(k) are called bonding,.. andantibonding ,... energy bands, re­spectively, and the plot is givenin units of Itl.

The overlap matrix Sij can be calculated by a similar method as used for

1iij, except that the intra-atomic integral yields the energy for the crystal Hamil­

tonian 1ii;, but the overlap matrix rather yields unity for the case of So;, if

we assume that the atomic wavefunction is normalized, SAA = SHn = 1 and

SAB = SBA = 2scos(kaj2), where s is the overlap integral between the nearest

A and B atoms,

(2.19)

The secular equation for the 2pz orbital of [(CH)z] is given by

I(2p - E 2(t - sE) cos(kaj2) I2(t - sE) cos(kaj2) (2p - E

(2.20)

«" - E)' - 4(' - ,E)' " ..'(k./2)0,

yielding the eigenvalues of the energy dispersion relations of Eq. (2.20) given by

E±(k) = (2p ±2tcos(kaj2), (_~ < k <~)1 ± 2scos(ka/2) a a

(2.21)

in which the + sign defines one branch and the - sign defines the other branch,

as shown in Fig. 2.2, where we use values for the parameters, (2p = 0, t = -1,

and s = 0.2. The levels E+ and E_ are degenerate at ka = ±,...E+(k) and E_(k) are called bonding,.. and antibonding 1f. energy bands,

respectively. Since there are two 1f electrons per unit cell, each with a different

2.3. TWO-DIMENSIONAL GRAPHITE 25

Fig. 2.3: (a) The unit celland (b) Brillouin zone of two-­dimensional graphite are shownas the dotted rhombus and theshaded hexagon, respecti vely. iii,and bi, (i = 1,2) are unit vec­tors and reciprocaJ lattice vec­tors, respectively. Energy disper­sion relations are obtained alongthe perimeter of the dotted tri­angle connecting the high sym­metry points, r, K and M.

spin orientation, both electrons occupy the bonding 1f energy band, which makes

the total energy lower than f2p.

2.3 Two-Dimensional Graphite

Graphite is a three-dimensional (3D) layered hexagonal lattice of carbon atoms.

A single layer of graphite, forms a two-dimensional (2D) material, called 2D

graphite or a graphene layer. Even in 3D graphite, the interaction between two

adjacent layers is small compared with intra-layer interactions, since the layer­

layer separation of 3.35A is much larger than nearest-neighbor distance between

two carbon atoms, ac_c=1.42A. Thus the electronic structure of 2D graphite is

a first approximation of that for 3D graphite.

In Fig. 2.3 we show (a) the unit cell and (b) the Brillouin zone of two­

dimensional graphite as a dotted rhombus and shaded hexagon, respectively,

where al and ii2 are unit vectors in real space, and bl and 62 are reciprocal

lattice vectors. In the :t, y coordinates shown in the Fig. 2.3, the real space

unit vectors at and ii, of the hexagonal lattice are expressed as

(2.22)

where a = liid = Iii, I = 1.42 x .j3 = 2.46A is the lattice constant of two­

dimensional graphite. Correspondingly the unit vectors 6\ and 6, of the recip--

26

roeal lattice are given by:

CHAPTER 2. TIGHT BINDING CALCULATION

b, =(~,_ 2<)V3a a

(2.23)

corr~ponding to a lattice constant of 4'11" /...I3a in reciprocal space. The directionof the unit vectors b1 and b'l of the reciprocal hexagonal lattice are rotated by!J00 from the unit vectors ill and il2 of the hexagonal lattice in real space, as

shown in Fig. 2.3. By selecting the first Brillouin zone as the shaded hexagon

shown in Fig. 2.3(b), the highest symmetry is obtained for the Brillouin zone

of 20 graphite. Here we define the three high symmetry points, r, K and Mas the center, the corner, and the center of the edge, respectively. The energy

dispersion relations are calculated for the triangle r M K shown by the dotted

lines in Fig. 2.3(b).As discussed in Sect. 2.3.2, three (J' bonds for 2D graphite hybridize in a

sp2 configuration, while, and the other 2pz orbital, which is perpendicular to

the graphene plane, makes 11" covalent bonds. In Sect. 2.3.1 we consider only 'II'"

energy bands for 2D graphite, because we know that the 11" energy bands are

covalent and are the most imporlant for determining the solid state properties

of graphite.

2.:1.1 11" Bands of Two-Dimensional Graphite

Two Bloch functions, constructed from atomic orbitals for the two inequivalent

carbon atoms at A and B in Fig. 2.3, provide the basis functions for 2D graphite.

When we consider only nearest-neighbor interactions, then there is only an in­

tegration over a single atom in '1lA A and 1f.BB, as is shown in Eq. (2.16), and

thus 'H.AA = ?iBB = (2p. For the off-diagonal matrix element 'HAB, we mustconsider the three nearest~neighborB atoms relative to an A atom, which are

denoted by the vectors HI.~' and R3 . We then consider the contribution to

Eq. (2.17) from Hl)i" and R3 as follows:

(2.24)'/(k)

2.3. TWO-DIMENSIONAL GRAPHITE 27

where t is given by Eq. (2.18)* and f(k} is a function of the sum of the phase

factors of eik.Aj (j = 1,···,3). Using the z,y coordinates of Fig. 2.3(a}, f(k} is

given by:

(2.25)

Since f(k} is a complex function, and the Hamiltonian forms a Hermitian ma­

trix, we write 1tBA = 1f.A.B in which ~ denotes the complex conjugate. Us­

ing Eq. (2.25), the overlap integral matrix is given by SAA= SBB =:: 1, and

SAB =:: sf(k) =:: SiJA' Here s has the same definition as in Eq. (2.19), so that

the explicit forms for Jl and S can be written as:

_ (", t/(k») _( 1 'f(k»)Jl_ , S_

t/(k)" '" '/(k)" 1(2.26)

Solving the secular equation det(Jl - ES) =:: 0 and using 1i and S as given in

Eq. (2.26), the eigenvalues E(t) are obtained as a function w(t), k", and k,:

E (k) _ '" ± tw(k)g2D - 1 ±sw(t) , (2.27)

where the + signs in the numerator and denominator go together giving the

bonding 11" energy band, and likewise for the - signs, which give the anti-bonding

1f~ band, while the function w(t} is given by:

w(k) = VI/(k)I' =..;3k",a k a k a

1 + 4 cos --- "" -'- +4 cos2 .:.I.: .2 2 2

(2.28)

In Fig. 2.4, the energy dispersion relations of two-dimensional graphite are

shown throughout the Brillouin zone and the inset shows the energy dispersion

relations along the high symmetry axes along the perimeter of the triangle shown

in Fig. 2.3(b). Here we use the parameters i2p =:: 0, t =:: -3.033eV, and s =:: 0.129

in order to reproduce the first principles calculation of the graphite energy bands

[9,48]. The upper half of the energy dispersion curves describes the 1I"~-energy

anti-bonding band, and the lower half is the 1I"-energy bonding band. The upper

1f~ band and the lower 1r band are degenerate at the K points through which

~We often use the symbol 'Yo for the nelU'ftt neighbor tranllfer integrl'l. 'Yo i. defined by &

pOIIitive value.

28 CHAPTER 2. TIGHT BINDING CALCULATION

"*r r--.---,----,15.0

>10.0

~ 5.0

"r M K

K

Fig. 2.4: The energy dispersion relations for 20 graphite areshown throughout the whole region of the Brillouin zone. Theinset shows the energy dispersion along the high symmetry direc­tions of the triangle rM K shown ill Fig. 2.3(b) (see text).

the Fermi energy passes. Since there are two 11" electrons per unit ceU, these two'II" electrons fully occupy the lower 'II" band. Since a detailed calculat.ion of the

density of states shows that the density of states at the Fermi level is zero, two­

dimensional graphite is a zero-gap semiconductor. The existence of a zero gap at

the f( points comes from the symmetry requirement that the two carhon sites A

and B in the hexagonal lattice are equivalent to each other. f The existence of a

zero gap at the J( points gives rise to quantum effects in the electronic structure

of carbon nanotubes, as shown in Chapter 3.

When the overlap integral s becomes zero, the "If and 1f" bands become

symmetrical around E = (2p which can be understood from Eq. (2.27). The

energy dispersion relations in the case of s = 0 (Le., in the Slater-Koster scheme)

are commonly used as a simple approximation for the electronic structure of a

graphene layer:

E,>D(k.,k,) =±t {I +4,~ (.j3~,"),~ (';") +4M' (k;")r(2.29)

I If the A and B sites had different atoms such all B and N, the site energy f'lp would bedifferent for B and N, and therefore the calculated energy dispersion would show an energygap between the,.- and,.-' bands.

2.3. TWO-DIMENSIONAL GRAPHITE 29

In this case, the energies have the values of ±3t, ±t and 0, respectively, at thehigh symmetry points, r, At and K in the Brillouin zone. Thus the band width

gives 16tl, which is consistent with the three connected 11" bonds. The simpleapproximation given by Eq. (2.29) is used in Sect. 4.1.2 to obtain a simpleapproximation for the eledronic dispersion relations for carbon nallotubes.

Finally let us consider the (J bands of two-dimensional graphite. There ace threeatomic orbitals of Sp2 covalent bonding per carbon atom, 25, 2p,. and 2py. Wethus have six Bloch orbitals in the 2 atom unit cell, yielding six u bands. We will

calculate these six (j bands using a 6 X 6 Han\iltonian and overlap matrix, andwe will then solve the secular equation· for each k point. For the eigenvalues

thus obtained, threeofthe six u bands are bonding u bands which appear belowtbe Fermi energy, and the other three u bands are 8ntibonding u· bands abovethe Fermi energy.

The calculation of the Hamiltonian and overlap matrix is performed ana.­

lytically, using a small number of parameters. Hereafter we arrange the matrix

elements in accordance with their atomic identity for the free atom: 28A, 2P1,

211, 2sB, 2p~, 2P:. Then the matri.x elements coupling the same atoms (forexample A and A) can be expressed by a 3 x 3 small matrix which is a sub-blockof the 6 x 6 matrix. Within the nearest neighbor site approximation given byEq. (2.16), the small Hamiltonian and overlap matrices are diagonal matrices as

follows,

'Ii" = (~,. ~" ~) SAA = (~ ~ ~) (2.30)00(2, 001

where (2p is defined by Eq. (2.16) and (2. is the orbital energy of the 28 levels.The matrix element for the Bloch orbitals between the A and B atoms can

be obtained by taking the components of 2p~ and 2py in the directions parallelor perpendicular to the (J bond. in Fig. 2.5, we show how to rotate the 2p,.

atomic orbital and how to obtain the (J and 'If components for the rightmost

·SinO!: the planar geometry of grAphite satiJies the even symmetry of the H.vniltonian 'Ii lUIdof '2,s, '2p" and 2p,. upon milTOr !'eRection about the r~ pla.oe, and the odd symmetry of 2p~, tbetI and ,.- enerv bands ean be IoIved ""paratdy, b«ause matnx demenu of different .ymmdrytypes do not couple in the HAmiltonian.

30 CHAPTER 2. TIGHT BINDING CALCULATION

Fig. 2.5: The rotation of 2p~.

The figure shows bow to projedthe tT and • components alongthe indicated bond starting withthe 2pa: orbital. This method isvalid only fot p orbitals.

~~(7)0. ~(S) ~o•.

Fig. 2.6: The band param­eters for (T bands. The fOUf

cases from (1) to (4) correspondto matrix elements having 000­

vanishing values and the remain­ing fOUf cases from (5) to (8) cor·respond La vanishing matrix ele­menLs.

boud of this figure. In Fig. 2.5 the waveCunction of 12p.,) is decomposed inlo itstT and r components as follows:

12p",) =cos'i12P..>+sin ~12P.. ). (2.31)

This type of decomposition can be used to describe a bond in any general direc­

tion, which is discussed in Sect. 1.2. This procedure is also useful for fullerenesand carbon nanolubes, when we consider the curvature of their surfaces.

By rotating the 2p., and 2P. orbitals in the directions parallel and perpen­

dicular to the desired bonds, the matrix elements appear in only 8 patterns as

shown in Fig. 2.6, where shaded and not-shaded regions denote positive and neg­

ative amplitudes of tbe wavefunctioDs, respectively. The four cases from (I) to

(4) in Fit;. 2.6 correspond to non-vanishing matrix elements and the remaining

four cases from (5) to (8) correspond to matrix elements which vanish because

of symmetry. The corresponding parameters for both the Hamiltonian and the

overlap matrix elements are shown in Fig. 2.6.

In Figs. 2.7(a) and (b) weshow examples of the matrix elements of (2sA 1?l12p:)

and (2": 1?l12~), respectively, obtained by the methods described above. Intbe case of Fig. 2.7 (a), there is only one non-vanishing contribution, and this

2.3. TWO-DIMENSIONAL GRAPHITE 31

Fig. 2.7: Examples of theHamilLonian matrix elements of(J' orbitals, (a) (2.AI1l12~) and(b) (2P:I1/12.:'). By w,.'ingthe 2p orbitals, we tel. the ma­trix elements in Eq. (2.32) andEq. (2.33), respectively.

(b)(alCXJ>

comes from Fig. 2.6 (2). The other pertinent cases of Fig. 2.6 (5) or (6) give

matrix elements that vanish by symmetry. Multiplying the phase factors for the

three nearest neighbor B atoms with the matrix elements, we get the following

result:

(2.32)

Similarly, in the case of Fig. 2.7 (b), the non-zero matrix elements correspond

to the cases of Fig. 2.6 (3) and (4),

= ~(1it1 +1t. )e-;i:~o/2,J3eit.O/2- JIl(1iv +1f")e-iltz·/'W3e-il:"a/2 (2.33)

£j!(1l" +1i. )e- i .l: .... / 2-.13 sin ¥.The resulting matrix element in Eq. (2.33) is a pure imaginary. However, thecalculated results for the energy eigenvalues give real values.

When all the matrix elements of the 6 x 6 Hamiltonian and overlap matrices

are calculated in a similar way, the 6 x 6 Hamiltonian matrix is obtained as a

function of I.:r; and kyo For given k points we then calculate the energy dispersionof the q bands from the secular equation of Eq. (2.14). The results thus obtained

for the calculated q and 'If energy bands are shown in Fig. 2.8. Here we have

used the parameters listed in Table 2.1, yielding a fit of the functional form of

the energy bands imposed by symmetry to the energy values obtained for the

first principles band calculations at the high symmetry points (48].

32 CHAPTER 2. TIGHT HINDING CALCULATION

Fig. 2.8: The energy dispersionrelations for (T and 1( bands oftwo-dimensional graphite. Herewe used the pa.ranlet.ers listed inTable 2.1.KM

20.0

10.05"...!;;

~0.0

-10.0

G

-20.0K r

value0.2120.1020.1460.129

s..S.,S.S" =s

1i.. 6.169?-l,p -5.580r(, -5.0371i. == t -3.033

Table 2.1: Values for t.he coupling parameters for carbon at.oms in the Hamilto­nian for :II" and (T bands in 2D graphite.

~ vaJue(eV) S

(-)The value for (2. is given relative La seLLing (2,. = O.

In the absence of more detailed experimental or theoretical information,these parameters can he used as a first approximatton in describing the matrix

elements for most sp' carbon materials fot which the carbon-carbon distance

is close to that of graphite, 1.42A. Further, if the parameters are only sligbtlychanged, the formulation can be used to describe the sri' diamond system andsp carbyne materials.

As is shown in Fig. 2.8, the 11" and the two u bands cross each other (i.e., have

different syrrunetries), as do the 11". and the two u· bands. However, becauseof the different group theoretical symmetries between u and 11" bands, no band

separation occurs at the crossing points. The relative positions of these Cf088ingsare known to be important for: (1) photo-transitions from tT to r· bands and

2.3. TWO·DfMENSIONAL GRAP1lITE 33

from If to (TO bands, which satisfy t.he selection rule for electric dipole transit.ions,

and (2) charge t.ransfer from alkali met.al ions to graphene sheets in graphite

intercalation compounds.

Using t.he basic concepuoftwo-dimensionalt;raphite presented in Chapter 2,

we next discuss the structure and electronic properties or sint;le-wall carbon

nanot.ubes in Chapters 3 and 4, respectively.